Re: Distilling water
Urine, blood, almost any organic liquid; I'm not guaranteeing flavour here, note, just lack of toxins. But the smoke of most combustion contains a fair amount of steam, too; what are they burning to boil the water? (Wonders which of camel dung or priceless tapestries is going to condense down to the least unpleasant tipple?) With a long enough tube through a cool shady grotto you might be able to collect enough to make a difference.
You can get by without ice (although by inefficient and expensive means I can make ice) but desert cities have cool regions for their citernas; evaporation is a problem even when the water is flowing normally. Still, it would be better to do the distillation at night; even in hot climates, desert nights get cold (lack of cloud to trap the heat, and humidity to regulate it).
I'm now considering some way of collecting sweat; absorbent inner garments that you wring out into the distillation tanks protected by waterproof overgarments?